In this problem, we return to the question of shrouded product attributes and
prices, introduced in Problem 6.14 and further analyzed in Problem \(14.13 .\)
Here we will pursue the question of whether market forces can be counted on to
attenuate consumer behavioral biases. In particular, whether competition and
advertising can serve to unshroud previously shrouded prices. We will study a
model inspired by Xavier Gabaix and David Laibson's influential article.
\(^{19}\) A population of consumers (normalize their mass to 1 ) have gross
surplus \(v\) for a homogeneous good produced by duopoly firms at constant
marginal and average cost \(c .\) Firms \(i=1,2\) simultaneously post prices
\(p_{i} .\) In addition to these posted prices, each firm \(i\) can add a shrouded
fee \(s_{i},\) which are anticipated by some consumers but not others. For
example, the fees could be for checked baggage associated with plane travel or
for not making a minimum monthly payment on a credit card. A fraction \(\alpha\)
are sophisticated consumers, who understand the equilibrium and anticipate
equilibrium shrouded fees. At a small inconvenience cost \(e\), they are able to
avoid the shrouded fee (packing only carry-ons in the airline example or being
sure to make the minimum monthly payment in the credit-card example). The
remaining fraction \(1-\alpha\) of consumers are myopic. They do not anticipate
shrouded fees, only considering posted prices in deciding from which firm to
buy. Their only way of avoiding the fee is void the entire transaction (saving
the total expenditure \(p_{i}+s_{i}\) but forgoing surplus \(v\) ). Suppose firms
choose posted prices simultaneously as in the Bertrand model.
a. Argue that in equilibrium, \(p_{i}^{*}+s_{i}^{*}=v\) (at least as long as \(e\)
is sufficiently small that firms do not try to induce sophisticated consumers
not to avoid the shrouded fee). Compute the Nash equilibrium posted prices
\(p_{i}^{*}\). (Hint:
As in the standard Bertrand game, an undercutting argument suggests that a
zero-profit condition is crucial in determining \(p_{i}^{*}\) here, too.) How do
the posted prices compare to cost? Are they guaranteed to be positive? How is
surplus allocated across consumers?
b. Can you give examples of real-world products that seem to be priced as in
part (a)?
c. Suppose that one of the firms, say firm \(2,\) can deviate to an advertising
strategy. Advertising has several effects. First, it converts myopic consumers
into sophisticated ones (who rationally forecast shrouded fees and who can
avoid them at cost \(e\) ). Second, it allows firm 2 to make both \(p_{2}\) and
\(s_{2}\) transparent to all types of consumers. Show that this deviation is
unprofitable if $$e<\left(\frac{1-\alpha}{\alpha}\right)(v-c)$$
d. Hence we have shown that even costless advertising need not result in
unshrouding. Explain the forces leading advertising to be an unprofitable
deviation.
e. Return to the case in part (a) with no advertising, but now suppose firms
cannot post negative prices. (One reason is that sophisticated consumers could
exact huge losses by purchasing an untold number of units to earn the negative
price, which are simply disposed of.) Compute the Nash equilibrium. How does
it compare to part (a)? Can firms earn positive profits?
